What Distinguishes Rings, Fields, and Spaces in Mathematics?

  • Thread starter Thread starter Gregg
  • Start date Start date
  • Tags Tags
    Fields Rings
AI Thread Summary
Rings, fields, and vector spaces are all mathematical structures defined by sets with specific addition and multiplication axioms. A field is a specialized type of ring where every nonzero element has a multiplicative inverse, making all fields rings but not all rings fields. Vector spaces require both a set of vectors and a field of scalars, distinguishing them from rings and fields. While fields can be viewed as vector spaces over themselves due to similar axioms, they remain fundamentally different structures. Understanding these distinctions is crucial for grasping advanced mathematical concepts.
Gregg
Messages
452
Reaction score
0
They all seem to be defined as sets with multiplication and addition axioms satisfied. What is the difference?
 
Mathematics news on Phys.org
A field is a ring where every nonzero element has a multiplicative inverse. All fields are rings, but not vice-versa. What spaces are you talking about, vector spaces?
 
Maybe spaces is not accurate but there seem to be a lot of things which are defined as having satisfying similar axioms.
 
Yes, it's true, but they do all have their differences. Vector spaces, for example, need both a set of vectors and a field of scalars. You can treat a field as a vector space over itself, because of the similarity of the axioms, but they are intrinsically different.
 

Thread 'Non-orthogonal bases'

Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
View full post »

Thread 'Fermat's Last Theorem'

Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
View full post »

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »

Similar threads

I ##C^{\infty}##-module of smooth vector fields can lack a basis
Replies
9
Views
3K
B Is everything in math either an axiom or a theorem?
2
Replies
72
Views
7K
A What if the (semi) field characteristic of N is not zero?
Replies
3
Views
2K
A Near-Rings with Noncommutative Addition and Two-Sided Distributivity
Replies
4
Views
2K
I Does a set of matrices form a ring? Or what is the algebraic structure?
Replies
24
Views
3K
I What Is a Torsion Element in Ring Theory?
Replies
13
Views
3K
Insights Why Vector Spaces Explain The World: A Historical Perspective
Replies
0
Views
7K
I Magnets, Outer space, and Rings
Replies
3
Views
1K
I What's the difference between a ring and a field?
Replies
9
Views
5K
I How to determine if a set is a semiring or a ring?
Replies
2
Views
3K

Hot Threads

Recent Insights

Back
Top